Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Besov--Triebel--Lizorkin-Type Spaces with Matrix $A_\infty$ Weights

Fan Bu, Tuomas Hytönen, Dachun Yang, Wen Yuan

TL;DR

This work develops a comprehensive theory for Besov-type and Triebel--Lizorkin-type spaces with matrix $A_{p,\infty}$ weights, extending the classical scalar $A_{\infty}$ framework to the matrix setting. It introduces inhomogeneous matrix-weighted spaces via averaging constructions, provides $\varphi$-transform characterizations, and proves molecular, wavelet, and Calderón--Zygmund-type decompositions under the $A_{p,\infty}$ regime. A key advancement is handling the lack of Fefferman--Stein inequalities in the matrix $A_{p,\infty}$ setting by leveraging averaging spaces and discrete Calderón reproducing formulae, enabling boundedness results for almost diagonal operators, trace and extension operators, and pointwise multipliers. The results include unweighted and scalar-weighted cases as special instances and offer sharp bounds that reflect the underlying $A_{p,\infty}$-dimensions, providing a robust framework for analysis of matrix-weighted function spaces with broad potential applications in harmonic analysis and PDEs.

Abstract

Introduced by A. Volberg, matrix $A_{p,\infty}$ weights provide a suitable generalization of Muckenhoupt $A_\infty$ weights from the classical theory. In our previous work, we established new characterizations of these weights. Here, we use these results to study inhomogeneous Besov-type and Triebel--Lizorkin-type spaces with such weights. In particular, we characterize these spaces, in terms of the $\varphi$-transform, molecules, and wavelets, and obtain the boundedness of almost diagonal operators, pseudo-differential operators, trace operators, pointwise multipliers, and Calderón--Zygmund operators on these spaces. This is the first systematic study of inhomogeneous Besov--Triebel--Lizorkin-type spaces with $A_{p,\infty}$-matrix weights, but some of the results are new even when specialized to the scalar unweighted case.

Besov--Triebel--Lizorkin-Type Spaces with Matrix $A_\infty$ Weights

TL;DR

This work develops a comprehensive theory for Besov-type and Triebel--Lizorkin-type spaces with matrix weights, extending the classical scalar framework to the matrix setting. It introduces inhomogeneous matrix-weighted spaces via averaging constructions, provides -transform characterizations, and proves molecular, wavelet, and Calderón--Zygmund-type decompositions under the regime. A key advancement is handling the lack of Fefferman--Stein inequalities in the matrix setting by leveraging averaging spaces and discrete Calderón reproducing formulae, enabling boundedness results for almost diagonal operators, trace and extension operators, and pointwise multipliers. The results include unweighted and scalar-weighted cases as special instances and offer sharp bounds that reflect the underlying -dimensions, providing a robust framework for analysis of matrix-weighted function spaces with broad potential applications in harmonic analysis and PDEs.

Abstract

Introduced by A. Volberg, matrix weights provide a suitable generalization of Muckenhoupt weights from the classical theory. In our previous work, we established new characterizations of these weights. Here, we use these results to study inhomogeneous Besov-type and Triebel--Lizorkin-type spaces with such weights. In particular, we characterize these spaces, in terms of the -transform, molecules, and wavelets, and obtain the boundedness of almost diagonal operators, pseudo-differential operators, trace operators, pointwise multipliers, and Calderón--Zygmund operators on these spaces. This is the first systematic study of inhomogeneous Besov--Triebel--Lizorkin-type spaces with -matrix weights, but some of the results are new even when specialized to the scalar unweighted case.
Paper Structure (22 sections, 89 theorems, 380 equations)

This paper contains 22 sections, 89 theorems, 380 equations.

Table of Contents

  1. Introduction
  2. Background on Matrix Weights
  3. Matrix $A_\infty$ Weights
  4. Upper and Lower Dimensions of $A_{p,\infty}$ Weights
  5. Matrix-Weighted Besov-Type and Triebel--Lizorkin-Type Spaces
  6. Preliminaries for Function Spaces
  7. Averaging Spaces $A^{s,\tau}_{p,q}(\mathbb{A})$ and $a^{s,\tau}_{p,q}(\mathbb A)$
  8. Pointwise Weighted Spaces $A^{s,\tau}_{p,q}(W)$ and $a^{s,\tau}_{p,q}(W)$
  9. Reproducing Formula with Generic Sampling
  10. Relation to Homogeneous Spaces
  11. Almost Diagonal Operators
  12. Homogeneous Sequence Spaces
  13. Comparison with Results of Bownik and Ho
  14. Inhomogeneous Sequence Spaces
  15. Molecular Characterization and Its Applications
  16. ...and 7 more sections

Key Result

Lemma 2.4

Let $p\in(0,\infty)$, $W$ be a matrix weight, and $E\subset\mathbb{R}^n$ a bounded measurable set satisfying $|E|\in(0,\infty)$. If $A_E$ is a reducing operator of order $p$ for $W$, then, for any matrix $M\in M_m(\mathbb{C})$, where the positive equivalence constants depend only on $m$ and $p$.

Theorems & Definitions (197)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 2.7
  • Proposition 2.8
  • proof
  • Proposition 2.9
  • ...and 187 more